Mirages in galaxy scaling relations

Transcription

1 on. Not. R. Astron. Soc. 000, 1 (14) Printed 12 October (N LATEX style file v2.2) Edge-on spiral galaxies provide a possibility to analyze the vertical surface brightness distribution and, therefore, add a new parameter to scaling relations the vertical scaleheight, typically, h z for the exponential vertical surface brightness distribution (Wainscoat, Freeman & HyarXiv: v1 [astro-ph.ga] 28 ar 14 irages in galaxy scaling relations A.V. osenkov 1,2, N.Ya. Sotnikova 1,3, and V.P. Reshetnikov 1,3 1 St.Petersburg State University, Universitetskij pr. 28, St.Petersburg, Stary Peterhof, Russia 2 Central Astronomical Observatory of RAS, Russia 3 Isaac Newton Institute of Chile, St.Petersburg Branch Accepted 14???. Received???; in original form 13??? 1 INTRODUCTION Global characteristics of galaxies (luminosity, size, rotational velocity, velocity dispersion, etc.) are not distributed randomly, but form a set of well-defined scaling relations. These relations are of great importance since they provide invaluable constraints on the formation scenarios and evolutionary processes of galaxies. The success of any particular theory will be judged by its ability to reproduce the slope, scatter, and zero-point of known scaling relations. One of the most firmly established empirical scaling relations of elliptical galaxies is the Fundamental Plane (FP) which represents a tight correlation between the surface brightness, the size, and the velocity dispersion of a galaxy (Djorgovski & Davis 1987; Dressler et al. 1987). Two different projections of the FP are also well-known: the Kormendy and the Faber Jackson relations (Kormendy 1977a,b; Faber & Jackson 1976). Spiral galaxies are more complex since they consist of a mosenkovav@gmail.com ABSTRACT We analyzed several basic correlations between structural parameters of galaxies. The data were taken from various samples in different passbands which are available in the literature. We discuss disc scaling relations as well as some debatable issues concerning the so-called Photometric Plane for s and elliptical galaxies in different forms and various versions of the famous Kormendy relation. We show that some of the correlations under discussion are artificial (selfcorrelations), while others truly reveal some new essential details of the structural properties of galaxies. Our main results are as follows: (1) At present, we can not conclude that faint stellar discs are, on average, more thin than discs in high surface brightness galaxies. The central surface brightness thickness correlation appears only as a consequence of the transparent exponential disc model to describe real galaxy discs. (2) The Photometric Plane appears to have no independent physical sense. Various forms of this plane are merely sophisticated versions of the Kormendy relation or of the self-relation involving the central surface brightness of a /elliptical galaxy and the Sérsic index n. (3) The Kormendy relation is a physical correlation presumably reflecting the difference in the origin of bright and faint ellipticals and s. We present arguments that involve creating artificial samples to prove our main idea. Key words: galaxies: kinematics and dynamics galaxies: structure. disc, a, and some other components like, for instance, a bar and a ring. The multi-component structure of spiral galaxies results in a variety of scaling relations involving parameters associated with a disc, a, and a galaxy as a whole. The most famous relation is, of course, the Tully Fisher law (Tully & Fisher 1977) which links galactic total luminosity and rotation velocity (usually taken as the maximum of the rotation curve well away from the center). A tight scaling relation may also exist between the photometric and kinematic characteristics of the discs alone. For instance, Karachentsev (1989), oriondo, Giovanelli & Haynes (1999), and others have discussed a threedimensional plane involving the disc scalelength h, the maximal rotational velocity v m, and the deprojected central surface brightness S 0,d. This plane is analogous to the FP of elliptical galaxies.

2 2 A.V. osenkov, N.Ya. Sotnikova, and V.P. Reshetnikov land 1989) or z 0 for the isothermal law (Spitzer 1942; van der Kruit & Searle 1981a,b, 1982a,b). A significant correlation between the central surface brightness of a stellar disc reduced to the face-on orientation S 0,d and the ratio h/z 0 was found: the thinner the galaxy, the fainter the central surface brightness (Bizyaev & itronova 02; Bizyaev & Kajsin 04; Bizyaev & itronova 09). Bulges of spiral galaxies also show several empirical relations. For instance, they follow a relation similar to the FP for elliptical galaxies (e.g. Falcón-Barroso, Peletier & Balcells 02). Khosroshahi et al. (00a) and Khosroshahi, Wadadekar & Kembhavi (00b) found a tight correlation between the Sérsic index n, the central surface brightness µ 0,b, and the effective radius of the r. They called this relationship the photometric plane (PhP). The PhP projection the correlation between the central surface brightness of a µ 0,b and its Sérsic index n is known as well (e.g. Khosroshahi, Wadadekar & Kembhavi 00b; öllenhoff & Heidt 01; Aguerri et al. 04; Ravikumar et al. 06; Barway et al. 09). Also, there exist mutual correlations between the structural parameters of discs and s (e.g. osenkov, Sotnikova & Reshetnikov 10, SR10 hereafter, and references therein). In this paper, we critically examine several important scaling relations of spiral and elliptical galaxies focusing on spirals, their discs and s. Our main conclusion is that some of these empirical relations (the deprojected central surface brightness the relative thickness of the disc, the central surface brightness of the the Sérsic index, the PhP for ellipticals and s of spirals) are not physical, and they merely reflect the structure of fitting formulas. In other words, these scaling relations are spurious self-correlations, or mirages of the approximation procedures. Such spurious self-correlations arise when two parameters, for example, A and B that are used in a linear regression analysis, have a common term: A = f(x) and B = f(x) + g(y), where x and y are random, uncorrelated variables. In this case, any correlation found between A and B has no physical meaning and is entirely due to the selfcorrelation associated with the shared variable x. Thus, selfcorrelations link a measured parameter A with an expression B including the same parameter. Examples are presented to show that under certain conditions perfect (but entirely spurious) correlation is obtained between two such parameters formed from random numbers. On the other hand, we show that the curvature of the Kormendy relation is real and can not be explained in terms of other linear relations unifying faint and bright galaxies as well as faint and bright s (Graham & Guzmán 03; Graham 11, 13a). This paper is organized as follows. In Section 2, we describe the samples analyzed in this paper. We briefly discuss the methods of deriving photometric parameters of s and discs. In Section 3, we discuss one well-known scaling relation for edge-on discs (between the central surface brightness of a stellar disc and the relative thickness) and show that it is a self-correlation. In Section 4, we demonstrate that the Photometric Plane for s, and ellipticals, and its various forms are merely the self-relation involving the central surface brightness of a /elliptical and the Sérsic index n or sophisticated versions of the Kormendy relation. In Section 5, we present arguments in favor of the reality of the Kormendy relation which do reveal important features of the galaxy structure. In Section 6, we summarize our main conclusions. 2 THE SAPLES We consider some of the most well-known samples with published decomposition results. The samples of edge-on galaxies are provided in Table 1. Other selected samples of galaxies are listed in Table 2. These samples comprise objects of different morphological types as well as are given in different photometric bands. Some samples are quite enormous (Simard et al. 11) or huge (e.g., Allen et al. 06 and Gadotti 09, hereafter G09) whereas others consist of only tens objects. We do not consider spheroidal galaxies and core elliptical galaxies since they are out of the scope of this article. It should be noted that the structural parameters for these samples were derived using various approaches. There are two basic methods: the one-dimensional (1D) and the two-dimensional (2D) methods. In the 1D method, the azimuthally averaged surface brightness profile of a studying galaxy, or major/minor axes profiles, are fitted by one or more components. This method has the advantage of being simple and fast and works in low signal-to-noise conditions. However, in 2D fitting, information from the whole image is used to build a more robust model for each component. There are several examples in the literature showing that the 2D method is much more reliable than the 1D method (e.g. de Jong 1996) retrieving more accurate structural parameters. In this article we do not compare these methods, but rather discuss the main results coming from all of them, regardless of the fitting procedure. We should note here that distances to galaxies used by the authors were differently estimated for each sample. The Hubble constant H 0 varies from 70 to 75 km s 1 pc 1 what may slightly change the distances. In addition to that, for some samples there was no information given on how those distances were found, e.g. were the radial velocities of the galaxies corrected to the centroid of the Local Group or to the galactic center. The vast majority of galaxies from the samples are not nearby, and, thus, such corrections do not change significantly the distances (in this case the difference may variate up to 10%). 3 DISCS: SCALING RELATIONS INVOLVING SCALEHEIGHT The disc structure out from the galaxy midplane can be investigated only for a special galaxy orientation when a disc galaxy is seen edge-on to the line of sight. It provides a unique opportunity to build a full 3D model of a galaxy and to define the disc thickness. Observations of the edge-on galaxies reveal also large-scale features that would otherwise remain hidden, like warps, truncations, bright halos, and boxy/peanut-shaped s. For objects thus oriented, one can study the distributions and ages of stellar populations. All these issues provide essential insights into the formation and evolution of disc galaxies.

3 irages in galaxy scaling relations 3 Table 1. List of analyzed samples of edge-on galaxies with derived structural parameters of discs. Reference Number of galaxies Band orphological types Bizyaev & itronova (02) (B02) 134 J, H, K s late types osenkov, Sotnikova & Reshetnikov (10) (SR10) 165 J all types 169 H all types 175 K s all types Table 2. List of some published samples of galaxies with derived /disc structural parameters of s. Reference Number of galaxies Band orphological types Caon, Capaccioli & D Onofrio (1993) 45 B E and S0 acarthur, Courteau & Holtzman (03) 121 B, V, R, H late types öllenhoff (04) 26 U, B, V, R, I all types Allen et al. (06) (GC) B all types Simard et al. (11) g, r all types Gadotti (09) (G09) 946 g, r, i all types cdonald et al. (11) 286 g, r, i, z, H all types, Virgo Guttiérrez et al. (04) 7 r all types, Coma In this section, we focus on the vertical structure of galactic discs and on one scaling relation that incorporates the thickness of the stellar disc and its deprojected central surface brightness (Bizyaev & itronova 02, 09; B02 and B09 hereafter). As we use the relative thickness of the disc (z 0/h), the difference in distances to galaxies from different samples does not affect the relations. 3.1 Scaling parameters for edge-on discs The breakthrough study of edge-on galaxies appeared in the 1980s when van der Kruit, Searle (e.g., 1981a; 1982a), and then other authors wrote several classical papers on the study of edge-on galaxies. Since that time, much progress has been made to investigate these objects (e.g., Reshetnikov & Combes 1997; de Grijs 1998; Kregel, van der Kruit & de Grijs 02; Pohlen et al. 00, 04; Yoachim & Dalcanton 06) and to summarize the main conclusions made from previous studies (e.g., SR10; van der Kruit & Freeman 11). Following these studies, we can derive the parameters of two major stellar components: a and a disc, where the disc can be described by the law which comprises the exponential radial scale as well as the heightscale (these are necessary for building the 3D surface brightness distribution of an observed galactic disc): I(r, z) = I(0, 0) r ( r ) h K1 sech 2 (z/z 0), (1) h where I(0, 0) is the disc central intensity, h is the radial scalelength, z 0 is the isothermal scaleheight of the disc (Spitzer 1942), and K 1 is the modified Bessel function of the first order. This formula is valid only in the case of a perfectly transparent disc. Unfortunately, the dust within galaxy discs can strongly attenuate the light not only from their discs but also from the embedded s. Dust lanes which are especially prominent in early type spiral galaxies (the flocculent dust content often resides also in late type spirals) may cover the significant part of the galactic disc what can be crucial to correctly determine the disc and structural parameters. This effect can be considerable even in the NIR bands. For instance, an edge-on galaxy NGC 891 has a dust lane that is very visible in the K s band (see Fig. 1). In addition, one of the difficulties we are faced with while Figure 1. Decomposition of the edge-on galaxy NGC 891 (2ASS, K s band). Images from top to bottom are the galaxy, the model, and the residual image. The dust lane is distinct on the residual image. The derived parameters of the disc are µ 0,d =15.8 mag arcsec 2, h=95.8 arcsec, z 0 =12.9 arcsec; for the : µ =17.7 mag arcsec 2, r =25.65 arcsec, n=2.3, and the apparent axis ratio q b =0.8. studying edge-on galaxies, is that we are not able to observe a spiral pattern in them. Thus, a guess as to the morphological type of a galaxy can be made mainly on the basis of its -to-disc luminosity ratio. 3.2 Central surface brightness thickness relation B02 analyzed a sample of late-type edge-on galaxies in the J, H and K s bands (see Table 1). They have noted a strong correlation between the central surface brightness of a stellar disc and the h/z 0 ratio. This means that the thinner a galaxy is, the lower its central surface brightness reduced to the face-on orientation S 0,d (we will designate the apparent central surface brightness of edge-on galaxies as µ 0,d ). The same correlation was confirmed for the stellar disc structural parameters corrected for internal extinction (Bizyaev & Kajsin 04; B09). Bizyaev & Kajsin (04) noted that this extinction correction is rather small (the median value for their sample is about 0.1 mag/arcsec 2 ). Fig. 2 demonstrates the S 0,d z 0/h correlation. B09 concluded that a very wide scatter of points in this correlation is due to the relatively low accuracy in the µ 0,d, z 0 and h and is also due to the non-constant value of the mass-to-light ratio (/L) for different galaxies.

4 4 A.V. osenkov, N.Ya. Sotnikova, and V.P. Reshetnikov Figure 2. Correlation between the relative thickness of a disc and its reduced central surface brightness in the K s band. Data were taken from B02. Black filled circles correspond to the more reliable subsample (designated as x in the table 1 from B02). 3.3 How does the relation S 0,d z 0/h reveal itself in other samples The largest sample of edge-on galaxies with derived structural parameters of discs and s is the SR10 sample (see Table 1). It comprises both early and late type objects. The fits-images were taken from 2ASS in all three bands (J, H and K s). The sample is incomplete according to the V/V max test, but the subsample of 92 galaxies with angular radius r > 60 arcsec appears to be complete. The program BUDDA (Bulge/Disc Decomposition Analysis; de Souza, Gadotti & dos Anjos 04) was applied for performing /disc decomposition. As we have all needed structural parameters, we can construct the same relation as in B02. Let us compare the sample by SR10 and the B02 sample. In Fig. 3 we plotted the distributions of the parameters z 0/h and µ 0,d in the K s band. The B02 sample comprises mainly late-type spiral galaxies. That is why the distributions over photometric parameters for this sample look slighlty different in comparison with our sample, but the mean values of both samples are similar. The median values and standard deviations for the SR10 sample are the following: For the B02 sample: z 0/h = 0.25 ± 0.11, µ 0,d = ± 0.69 mag arcsec 2, z 0/h = 0.23 ± 0.07, µ 0,d = ± 0.56 mag arcsec 2. From these distributions we can see that the scatters of both parameters are relatively narrow, and the characteristics of the samples are close. Figure 3. Distributions of the relative thickness and the deprojected central surface brighness of the discs in the K s band for the sample by SR10 (top plots) and for the B02 sample (bottom plots). In Fig. 4 we show the mutual distribution of µ 0,d and z 0/h for our sample (row a, left plot) and for the B02 sample (row b, left plot). Right plots in Fig. 4 represent the S 0,d z 0/h correlation for our sample (row a) and for the B02 sample (row b). The regression line for the SR10 sample is S 0,d = 5.09 log(z 0/h) , r = 0.49, (2) and for the B02 sample is S 0,d = 5.17 log(z 0/h) , r = (3) Correlations for both samples are similar and rather strong, but are they real? It is known from the surface photometry of transparent discs that the central surface brightness of the face-on disc (when the inclination angle is i = 0 ) expressed in magnitudes per arcsec 2, can be reduced from the edge-on (apparent) central surface brightness as follows: S 0,d = µ 0,d 2.5 log(z 0/h). (4) From this expression (4) we may notice several useful facts. First, the scatter of S 0,d should be larger than that of µ 0,d because of the presence of the term log(z 0/h). Second, from (4) we can see that if z 0/h const, then there is a simple linear dependence between S 0,d and µ 0,d. Third, contrary, if µ 0,d const, there is a simple logarithmic dependence between S 0,d and z 0/h. Hence, the small scatters around the median values µ 0,d and z 0/h may transform the reduction formula (4) into the self-correlation between S 0,d and z 0/h because the expression for S 0,d contains the term of z 0/h. To prove this conclusion, we designed some examples. They show that under certain conditions, perfect (but entirely spurious) correlation is obtained between two parameters formed from random distributions. 3.4 Self-relation between central surface brightness and thickness: artificial samples We generated several samples of artificial galaxies with normal distributions of observed parameters µ 0,d and z 0/h. Although the distributions of µ 0,d and z 0/h are not normally

5 irages in galaxy scaling relations 5 distributed in reality (see Fig. 3), we use this simplification merely to show that the resultant correlation will be the same as that for the real data. The sample #1 (filled circles in Fig. 4c, left plot) is built to imitate the real distribution similar to our and the B02 samples with the following mean value of µ 0,d and its standard σ: µ 0,d = 16.5, σ = 0.6 mag arcsec 2. The sample #2 (open circles in Fig. 4c, left plot) has a wider distribution over µ 0,d : µ 0,d = 16.5, σ = 1.1 mag arcsec 2. In both cases the distribution of z 0/h was the same: z 0/h = 0.25, σ = We converted µ 0,d into S 0,d according to (4) and plotted the relation S 0,d log(z 0/h) (see the right column in Fig. 4). It appears to be linear with a scattering that is due to the scatter of µ 0,d and z 0/h. The regression line for the sample #1 is S 0,d = 3.96 log(z 0/h) , r = 0.545, (5) and the regression line for the sample #2 (with a wide distribution over µ 0,d ) is S 0,d = 1.41 log(z 0/h) , r = (6) The regression coefficient is much smaller for this broader distribution of µ 0,d (the sample #2, Fig. 4c, right plot, open circles, dashed line). Thus, the correlation S 0,d z 0/h for this sample is statistically insignificant. But for the artificial sample #1 containing random (uncorrelated) distributions of µ 0,d and z 0/h the regression coefficient and the slope of the correlation S 0,d z 0/h (Fig. 4c, right plot, filled circles, solid line) are almost the same as that for the B02 and SR10 samples. Thus, we can see that even if we have no correlation between the visible surface brightness of the edge-on disc and its relative thickness, there would be, nevertheless, the correlation between the reduced central surface brightness and the relative thickness of the disc. This correlation, however, can be substantially smoothed and even destroyed if the scatter of µ 0,d is rather large. To demonstrate this fact, we constructed two additional artificial samples. The sample #3 (filled circles in Fig. 4d, left plot) has a very small scatter of the ratio z 0/h (σ = 0.05) and a large scatter of µ 0,d (as for the sample #1) with the same mean values. The sample #4 (open circles in Fig. 4d, left plot), contrary, has a very small scatter of the value µ 0,d (σ = 0.2) and a wide distribution over the ratio z 0/h (σ = 0.1). As a consequence, the sample #3 do not show any correlation between S 0,d and z 0/h (filled circles in Fig. 4d, right plot). In other words, if there is a narrow scatter of z 0/h with the same distribution of µ 0,d as for the sample #1, then the S 0,d z 0/h correlation does not appear. On the contrary, if we have a large scatter of z 0/h with a very narrow distribution of µ 0,d, then the expected correlation will be very strong (open circles in Fig. 4d, right plot). Trying to explain the S 0,d z 0/h correlation, Bizyaev & Kajsin (04) noted that the values of S 0,d and z 0/h had not been obtained independently from each other as can be concluded from the Eq. (4). They considered several effects that can affect the correlation S 0,d z 0/h. In particular, they argue that a non 90 degree inclination of the disc simply shifts the data points in Fig. 4 towards the upper right corner because of the overestimation of z 0. Hence, systematic errors due to inclination may only scatter the dependence shown in Fig. 4 and do not affect a correlation if it exists. This explanation can not be adopted because there is certainly a scatter around the relation (4) with the median value of µ 0,d. The slope of the regression lines (2) and (3) is twice as large as the slope of the relation (4) with the median value µ 0,d, but the slope of the relation for artificial sample #1 is also larger (see the expression (5)). In other words, we can not assert the existence of the S 0,d z 0/h correlation beyond the self-correlation due to the reduction procedure (4). 3.5 Are there physical bases of the correlation S 0,d z 0/h? Let us turn to the possible explanation of the correlation S 0,d z 0/h, if it exists. Following B09 (see also Zasov et al. 02; Kregel, van der Kruit & Freeman 05; Sotnikova & Rodionov 06), we will consider the exponential disc which is in equilibrium in the vertical direction. For such a disc we can find the vertical scaleheight z 0 via the vertical equilibrium condition for an isothermal slab (Spitzer 1942): σz 2 = πgσz 0, where σ z is the vertical velocity dispersion, and Σ is the surface density of a slab. To express the central surface density through the central surface brightness, we can write: Σ S 0,d. The mass of the disc can be estimated as d = 2πΣ 0h 2. At R 2h, the rotation curves of luminous galaxies generally reach a plateau. In the plateau region, the linear circular velocity v c is roughly constant. We can then use v c to estimate the total mass of a galaxy (including the mass of its spherical component: +dark halo) within the sphere of the radius R = 4h: tot(4h) = 4vc 2 h/g. Thus, we have: tot(4h) d v2 c h 100.4S 0,d. (7) We can link the relative mass of a disc with its relative thickness via stability conditions. If the stellar disc is marginally stable in its plane, then the radial velocity dispersion can be written as σ R(R) = Q, where Q 3.36G Σ(R) κ(r) is the Toomre parameter (Toomre 1964), κ is the epicyclic frequency, R is the radius in the cylindrical reference frame associated with the disc. For marginally stable discs the radial profile of Q usually has a wide minimum with the value Q 1.5 in the region of (1 2) h. This value is justified by the results of numerical experiments by Khoperskov et al. (03). Thus, we can consider Q to be almost constant with the radius outside the disc center. The epicyclic frequency at the region where v c const, is κ 2v c/r. In summary, we obtain for 1 R 2h (see Sotnikova & Rodionov 06 for details): h 1 vc 2 /h z 0 (σ z/σ R) 2 Σ 1 tot(4h) (σ z/σ R) 2. (8) d 1 The reference distance of R 2h is chosen because Q and v c are almost constant there.

6 6 A.V. osenkov, N.Ya. Sotnikova, and V.P. Reshetnikov Figure 4. Correlations between the relative thickness and the central surface brightness (K s band): apparent (left column) and reduced (right column); a) the sample by SR10; b) the B02 sample; c) artificial samples #1 and #2 (filled circles and open circles respectively); d) artificial samples #3 (filled circles) and #4 (open circles), see the text. Solid lines correspond to the regression lines for the samples mentioned in the text. Dashed line is a regression line for the sample #2. If the ratio of vertical to radial velocity dispersions σ z/σ R is almost constant throughout the disc, we have a correlation between z 0/h and d / tot. The existence of such a correlation was for the first time mentioned by Zasov, akarov & ikhailova (02), and it was further explored in many works (e.g. Zasov et al. 02; Kregel, van der Kruit & Freeman 05; Sotnikova & Rodionov 06; SR10) and references in B09). The ratio σ z/σ R could be fixed at the level given by the local linear criterion for the marginal bending stability, i.e. at approximately 0.3 (Toomre 1966; Kulsrud et al. 1971; Polyachenko & Shukhman 1977; Araki 1985). Recent numerical experiments by Rodionov & Sotnikova (13) support this minimal value throughout the disc. For real galaxies, some mechanisms heating the disc in the vertical direction and causing an increase in the ratio σ z/σ R may operate. At present, the ratio σ z/σ R is measured directly only in a few galaxies (Gerssen et al. 1997, 00; Shapiro et al. 03; Gerssen & Shapiro Griffin 12). It ranges from 0.3 to 0.8, but for our purposes we can fix this value at any level. Now, combining (7) and (8) we can expect: h z 0 v2 c h 100.4S 0,d. (9) B09 came to a similar conclusion. They used the correlation h v 1.5 which they had observed, and found z 0/h Σ 0/h 1/3, where Σ 0 is the central surface density of a disc. They considered such a result to be the theoretical basis for their correlation between h/z 0 and S 0,d (B02). We need to emphasize, however, that in the relation (9) we have not only the term S 0,d but also vc 2 /h (in the B09 version it is h 1/3 ). From this correlation it is not evident that there is a correlation between S 0,d and (z 0/h) only! We may only conclude that there may be a correlation (9)

7 irages in galaxy scaling relations 7 is a simple generalization of r 1/4 (de Vaucouleurs 1948, 1953, 1959) and exponential laws by Freeman (1970) (see, for example, Davies et al. 1988; Young & Currie 1994; Graham 01 and references therein). The r 1/n profile is given by the formula: Figure 5. Correlation between z0/h and v 2 c /h S 0,d (which is proportional to d / tot) in the K s band. Open circles represent the complete subsample of the sample by SR10, filled circles represent the B02 subsample. The rotational velocity values v c were taken from the LEDA database (as uncorrected for inclination vmax output parameters supposing that the inclination i 90 for all galaxies considered). which actually takes place as we can see in Fig. 5. Correlation (9), however, comprises the S 0,d term which was received via relation (4). Correlation (9), therefore, exists in the same sense as the correlation S 0,d z 0/h exists. We can not prove the reality of this correlation for galaxies at moderate inclination for which S 0,d can be derived directly without reduction formula. Unfortunately, for such galaxies the ratio z 0/h is undefined. oreover, the correlation between z 0/h and d / tot that was used to come to (9), is rather ambiguous, mainly due to the term (σ z/σ R) 2 in the expression (8) (Sotnikova & Rodionov 06; SR10). It exists only in the sense that discs embedded into very massive halos are always very thin. 3.6 Conclusion The correlation between h/z 0 and S 0,d, if it exists, is rather weak and can not be derived from observational data because the main effect seen in Fig. 4 is predominantly due to data reduction, many assumptions, and specific mathematical laws used to describe disc surface brightness distribution. All together, it gives a predictable result, i.e. a self-correlation. 4 BULGES AND ELLIPTICALS: PHOTOETRIC PLANE 4.1 Background The overall shape of elliptical, dwarf elliptical and profiles can be quantified and parametrized by means of r 1/n law (Sérsic 1968) for the radial surface brightness I(r) which I(r) = I 0 e νn(r/re)1/n, (10) where r e is the effective radius, i.e. the radius of the isophote that contains 50% of the total luminosity of a galaxy or a, I 0 is the central surface brightness, n is the Sérsic index defining the shape of the profile, and the parameter ν n ensures that r e is the half-light radius. In magnitudes per arcsec 2 the expression (10) appears as follows ( ) 1/n 2.5 νn r µ(r) = µ 0 +, (11) ln 10 where µ 0 is the central surface brightness expressed in mag per arcsec 2. The coefficient ν n depends on n and is an almost linear function of the Sérsic index n. As usual, one implies a numerical approximation of n in any appropriate form. One of these approximatione which is valid in the range 0.5 n 16.5, is (Caon, Capaccioli & D Onofrio 1993) ν n n (12) ln 10 The profile of an elliptical galaxy (and a of a spiral) that is fitted with the Sérsic model, can be also expressed as ] µ(r) = µ e ν n [(r/r e) 1/n 1. (13) where µ e is the effective surface brightness, i.e. the surface brightness at r e. For the fitting purpose, we can use the formula (11) and consider µ 0 and n as free (fit) parameters fixing the range of possible values of µ e (e.g. Caon, Capaccioli & D Onofrio 1993). In this case the uncertainty associated with the determination of µ e arises because µ e = µ ν n can differ from its measured value µ e. The value of µ e can be further compared with the measured counterpart µ e to test the goodness of a fit. On the contrary, if a fit for a sample is ambiguous and comprises systematic errors, such errors may affect scaling relations. In the -disc decomposition, we have the following as free (fit) scaling parameters for a : (1) the central intensity I 0,b in counts what can be later converted to µ 0,b in mag arcsec 2 ; (2) the half-light radius of the r e in pixels; (3) the shape parameter n (e.g. Khosroshahi, Wadadekar & Kembhavi 00b; Khosroshahi et al. 04). In this case, µ can be calculated from the expression r e µ = µ 0,b ν n. (14) It has become customary to choose µ as a fit scaling parameter instead of µ 0,b (öllenhoff & Heidt 01; acarthur, Courteau & Holtzman 03; Balcells, Graham & Domínguez-Palmero 06; éndez-abreu et al. 10; G09; SR10). In this case, µ 0,b is not an independent parameter but is calculated from the formula (14) that involves n. 4.2 Photometric Plane as a bivariate relation The derived scaling parameters of galaxies may correlate. Correlations comprising the scaling parameters of Sérsic

8 8 A.V. osenkov, N.Ya. Sotnikova, and V.P. Reshetnikov models, are widely discussed in the literature as well as the physical reasons of such correlations. Graham & Guzmán (03) discussed several linear scaling relations for elliptical galaxies (mainly for des and intermediate to bright E galaxies). There are also bivariate correlations. One of them was introduced by Khosroshahi et al. (00a) and Khosroshahi, Wadadekar & Kembhavi (00b) and was called Photometric Plane (PhP). any authors have confirmed it for their samples of elliptical galaxies and s of spiral galaxies of all types in various bands (öllenhoff & Heidt 01; Ravikumar et al. 06; éndez-abreu et al. 10; Laurikainen et al. 10), in different environments (Khosroshahi et al. 04), and for faint and bright objects (Barway et al. 09). Khosroshahi, Wadadekar & Kembhavi (00b) presented the PhP as a bivariate relation that links only photometric parameters obtained by fitting a Sérsic model to a galaxy image (or to a ), i.e. the Sérsic index n, the central surface brightness 2 µ 0,b, and the effective radius of a galaxy, or of a r. For any sample we can perform the least-squared fit of the expression log(n) = a log(r ) + b µ 0,b + c (15) and find a, b and c 3. In the literature there are different versions of the Photometric Plane, and we refer to the plane in the form (15) as the PhP1. Khosroshahi, Wadadekar & Kembhavi (00b) concluded that there exist two univariate correlations between the effective radius and the Sérsic index n, and between n and the central surface brightness. These univariate correlations have a scatter that may be caused by a third parameter. The methods of multivariate statistics applied to the three parameters n, µ 0,b, and r may reduce the scatter and give the best-fit plane like that expressed by Eq. (15). The Photometric Plane is thought to be a counterpart of a plane of a constant specific entropy of galaxies introduced by Lima Neto et al. (1999). Lima Neto et al. (1999) proposed two laws that elliptical galaxies and s of spirals must obey if they form and reach a quasi-equilibrium stage solely under the influence of gravitational processes. The first law is the virial theorem, and the second one is that a system in equilibrium is in a maximum entropy configuration. árquez et al. (00, 01) argued that after violent relaxation spherical systems may be considered to be in a quasi-equilibrium stage. In this stage, the two above mentioned laws are valid, and they lead to quasi-constant specific entropy. Ravikumar et al. (06) expressed the specific entropy S in a convenient form via three photometric parameters µ 0,b, r in kpc, and n. If S = const, there exists the relation that connects only µ 0,b, r in kpc, and n. This relation gives the surface (plane) of a constant specific entropy. The value of specific entropy may be adjusted so to match the specific entropy plane with the Photometric Plane (see, for example, Khosroshahi et al. 04; Ravikumar et al. 06). Such a coincidence between two planes is thought to give a physical interpretation of the PhP1. The PhP1 2 Hereafter we denote the surface brightness for ellipticals and s as µ 0,b or µ to distinguish it from the surface brightness of discs. 3 Here and below we use lm function in R language to calculate coefficients of the model. may be understood as a consequence of the two laws mentioned above. A physical interpretation of the PhP1 given by árquez et al. (01), clarifies the processes that drove the formation and evolution of galaxies and proves that the PhP1 is not simply an artifact of the definitions of the photometric parameters. 4.3 Photometric Plane 1: is it flat? In previous papers (SR10; Sotnikova, Reshetnikov & osenkov 12), we revealed that the PhP1 in J, H and K s bands appeared not to be flat. It has a prominent curvature towards small values of n (with log(n) < 0.2). Such a curvature is not seen in early papers that used small samples with rather large values of n (Khosroshahi et al. 00a; Khosroshahi, Wadadekar & Kembhavi 00b; öllenhoff & Heidt 01), but it was noticed later (Khosroshahi et al. 04; Ravikumar et al. 06; Barway et al. 09) and discussed in terms of a curved specific entropy surface. We consider the reason for this curvature to be quite a different one, and it helps to understand the origin of the relation (15). To clarify the question, we found the coefficients of the expression (15) and constructed the PhP1 (Fig. 6a) in the B band for more than galaxies of all types from the illennium Galaxy Catalogue GC (Allen et al. 06) 4. We superimposed on this plane 45 intermediate to bright E galaxies from Caon, Capaccioli & D Onofrio (1993). All these samples contain values of µ 0, either fitted or recalculated from the model. We also added two samples containing structural parameters for s in the B band: 121 face-on galaxies of late types from acarthur, Courteau & Holtzman (03) and 26 non-barred bright galaxies of all types from öllenhoff (04). We did not consider dwarf galaxies because their structure can differ substantially from the structure of bright ellipticals and s. At the same time, we consider s and elliptical galaxies all together. We will superimpose them often on the same plots keeping in mind that these are physically different objects. We are inetersted here only in the studying the shape of constructed correlations and dependencies which, as we could make sure, are similar for both elliptical galaxies and s from different samples (despite of their possible shift relative to each other on the plotted graphs). We have mentioned that authors used different methods to calculate the distances to galaxies. The difference in distances may variate up to 10% and results in a slight difference of physical size of a galaxy. But when we compare different samples, we will consider the logarithm of the physical size of a component(for example, the effective radius of the ), so the scatter of its values for all the samples will be less in this scale and would not affect the shape of the relation. We constructed the same relation (Fig. 6b) in the r band for the sample from Simard et al. (11). The data for this sample comes from the Legacy area of the Sloan Digital Sky Survey Data Release Seven. This sample contains more than 4 We use the catalogue of structural parameters mgc gim2d.par from jliske/mgc/ recommended by authors. We select galaxies with total model magnitude m(b) < 19 mag and r > 0.1 kpc.

9 irages in galaxy scaling relations Univariate relations As was noted above, the bivariate correlation helps to diminish the scatter in univariate correlations. In our case, they are correlations between the Sérsic index n and the central surface brightness, and between the effective radius and n. It is important to stress that a narrow plane connecting photometric parameters reveals itself only if the expression (15) comprises the central surface brightness µ 0,b. A corresponding plane does not appear if one uses µ instead of µ 0,b (see, for example, öllenhoff & Heidt 01). Let us now consider two mentioned univariate relations Central surface brightness vs Sérsic index for s and ellipticals Figure 6. The Photometric Plane (PhP1) constructed for a) the GC sample (Allen et al. 06) in the B band; b) the subsample from Simard et al. (11) in the r band. Some other samples were superimposed on these planes (see the text and the legend). 1 million galaxies, sometimes very distant to be analyzed. Therefore we selected objects only with 0.02 z 0.07 (more than galaxies). To avoid the presence of too many data points on our plots, we randomly selected galaxies from the subsample. We also used a large sample of spiral and elliptical galaxies (946 objects) built by G09 where galaxies were originally selected with the same restriction on z. We added the sample of galaxies from the Virgo cluster (286 galaxies, cdonald et al. 11) and 7 galaxies in a region of the Coma cluster (mainly Coma cluster members, Guttiérrez et al. 04). We added to these data 43 galaxies with known structural parameters from acarthur, Courteau & Holtzman (03) and galaxies from öllenhoff (04). All samples used in our analysis are listed in Table 2. It turns out that for all samples there is a fairly tight correlation for s with n 2 (classical s) and ellipticals, and a big scatter of points for s with n 2 (pseudos). The curvature of the PhP1 is also quite visible. The reason for the curvature may lie in the different nature of objects with n 2 and n 2, or in something else. Graham & Guzmán (03) presented a tight linear relation between µ 0 and log(n) (their figure 9f). The data have been compiled from several samples of elliptical galaxies (Caon, Capaccioli & D Onofrio 1993; Binggeli & Jerjen 1998; Stiavelli et al. 01; Graham & Guzmán 03) with derived structural parameters in the B band (the Sérsic model was used). Such a correlation was noted as a very strong while analyzing data for elliptical dwarfs in the Coma cluster (Binggeli & Jerjen 1998; Kourkchi et al. 12). odeling the s of spiral galaxies, other authors have found a similar trend (e.g. Khosroshahi, Wadadekar & Kembhavi 00b; öllenhoff & Heidt 01; Aguerri et al. 04; Ravikumar et al. 06; Barway et al. 09). To explain this trend for des, Graham (11, 13a) discussed two key empirical linear relations from which the linear relation between µ 0 and log(n) can be derived. They are the luminosity-concentration (L n) relation and the luminosity-central density (L µ 0) relation which unify faint and bright elliptical galaxies along one linear sequence. This issue will be discussed in Section 5. It should be noted that the points in figure 9f from Graham & Guzmán (03) do go along a straight line, and the scatter looks natural because of inhomogeneity in the compiled sample and uncertainties while fitting photometric profiles. The deviation from the straight line is seen only at small values of n, but the sample in this range is poor (see also Graham 11, figure 2b). We reproduced the relation µ 0 log(n) in the B band (Fig. 7a) and in the r band (Fig. 7b) for all samples as in Fig. 6. The line µ 0 = log(n) in Fig. 7a was drawn as in Graham (11) where it has been estimated by eye. We also reproduced the relation µ 0 log(n) in the r band separately for the sample by G09 in the Fig. while discussing the Kormendy relation. For both bands the relation µ 0 log(n) clearly curves towards the range of small values of n and does not follow a straight line Discussion and explanation Several questions arise. Why is this relation curved? Is it real? Does it reflect some common physical processes which make spherical galaxies and s acquire their structure? Or, on the contrary, can this relation be explained simply by the procedure of image decomposition and surface brightness profile fitting?

10 10 A.V. osenkov, N.Ya. Sotnikova, and V.P. Reshetnikov Figure 7. The central surface brightness µ 0,b of the underlying host galaxy or of the elliptical shown against the Sérsic index n on a logarithmic axis; a) the data are in the B band where the solid line corresponds to the relation µ 0 = log(n) from Graham (11); b) the data were taken in the r band (the data of G09 are reproduced separetly in Fig. and in Fig., where they are plotted with n instead of log(n)). The samples used are listed in Table 2. To sort out these questions, let us first consider the relation µ vs n. Surprisingly, being primarily measured (as for ellipticals) or fitted (as for s), µ shows no trend with n. The top plot in Fig. 8 demonstrates data from several samples of elliptical galaxies and of spiral galaxy s in the B band. The compiled sample is inhomogeneous; the scatter is large. Some points fall off the main distribution. This is a case of bright galaxies by öllenhoff (04). But both for the entire sample and for each subsample we can not observe the trend. The lack of the trend clearly manifests itself for the largest sample of galaxies from GC. The sample is poorly inhabited in the region of large values of n, but the general behaviour is unambiguous. The straight line shows the median value of µ for GC s galaxies. Bright galaxies from öllenhoff (04) are above this line contributing only to the scatter, but without creating the trend. The most impressive example is shown in Fig. 8b. It shows the data in the r band for the s from the sample by Simard et al. (11). The data were complemented Figure 8. The effective surface brightness µ of the underlying host galaxy or of the elliptical shown against the Sérsic indices n; a) the data are in the B band, µ = mag arcsec 2 corresponds to the GC sample; b) the data were taken in the r band, µ =.48 mag arcsec 2 corresponds to the Simard et al. (11) subsample. In order to show the lack of trend of µ with n for the Simard et al. (11) subsample, the values of µ were avaraged inside the bin n = 0.5. The corresponding bars represent the standard deviation of µ inside each bin. The samples used are listed in Table 2. by several additional samples with available decompositions in the same r band as in Fig. 7b. In Fig. 8b the data from the very homogeneous sample of spiral galaxies by G09 are also plotted. It is clearly seen that the points merely scatter around the median value of µ (straight line 5 ). There is no trend of µ with n. There is a slight bend around the median value of µ for Simard s (11) data. The procedure of µ deriving is not direct for this sample, and the fitting procedure itself can be a reason of existence of that bend 6. Faint Virgo cluster galaxies (cdonald et al. 11) 5 The median value of µ was calculated for galaxies from the sample by Simard et al. (11). 6 For this sample the free fitting parameters were the total flux, the fraction B/T, the effective radius r, and the Sérsic index n. The values of µ and µ 0,b should be calculated through these parameters via appropriate formulas.

11 irages in galaxy scaling relations 11 and bright galaxies from öllenhoff (04) deviate from the straight line lying above and below the median line. The lack of the trend of µ with n was neither noted nor discussed earlier but helps us to understand the relation between µ 0 and n. As ν n is an almost linear function of n (see Eq. 12), µ 0,b can be expressed as: µ 0,b µ (2n 0.33), (16) where µ is the median value for a sample. This is a linear self-relation between µ 0,b and n. As ν n is an almost linear function of n (see Eq. (12)), we have a linear self-relation between µ 0,b and n (see Fig. for the sample by G09) that transforms into a curved selfrelation between µ 0,b and log(n) (see Fig. 7 and Fig. ) The relation between central surface brightness and Sérsic index: the Gadotti s sample To prove the above conclusion, we analyzed carefully the fiducial sample by G09 (we will often address to this sample further and use the data from the r band). This sample contains a large amount of objects which were selected and decomposed very carefully. Thus, the data can be considered as quite homogeneous. Here we present the distributions over fitted photometric parameters r, µ, and n for this sample, both for s and ellipticals. We divided the s into two subsamples (faint s with 19.3 and bright s with < 19.3) and considered separetely elliptical galaxies for which the -to-total ratio B/T = 1. The distributions are shown in Fig. 9. The reasons for the division into subsamples were as follows. G09 showed that classical s (n 2), pseudos (n 2), and bright elliptical galaxies are separate groups of objects. The most significant parameter separating these objects, is the Sérsic index n, and we can use boundary values of Sérsic model parameters distribution for several populations of objects. For the G09 sample there are two well visible peakes in distributions of r and n (see Fig. 9). At the same time, the effective surface brightness distributions are similar for pseudos and classical s, and we can not distinguish the peaks of both distributions. We took the boundary values r 0.9 kpc, µ.2 mag arcsec 2 (median value for the subsample of s and pseudos), and n 2.5. Then we put these values into the relation (see, for example, Caon, Capaccioli & D Onofrio 1994; Graham & Colless 1997; Graham & Driver 05) = µ 2.5 log(ne νn Γ(2 n)/ν 2 n n ) 2.5 log(2πr 2 ) 36.57, (17) As a result, we received sep 19.3 mag for the G09 sample. Fig. 9 demonstrates three distinct populations of objects. The middle plot in Fig. 9 shows the distributions over µ. The overall range of µ is rather small, no greater than 3 mag, but for each subsample the scatter is much smaller (about 1 mag). Thus, the distribution of µ gives only the scatter around the relation (16) (see Fig. 10 which will be discussed below). For small samples, the scatter around the relation between µ 0,b and n is small because of the limited range of µ (as for subsamples of faint and bright s). The wider the distribution over µ and the more inhomogeneous the compiled sample, the thicker the lane surrounding the relation (16) is, but the relation itself does not sink in the scatter. In summary, there is no linear correlation between µ 0,b and log(n). There is just an equality (14) which reflects the structure of the Sérsic model. The limited range of µ for any sample transforms this equality into the linear pseudorelation between µ 0,b and n (see Eq. (16)) creating a false illusion of a correlation, i.e a self-correlation. oreover, at the limited range of n any linear relation can be presented as logarithmic, i.e. depending on log(n). That is why there is no mystery in the widely discussed relation µ 0,b vs log(n) (Binggeli & Jerjen 1998; Khosroshahi et al. 00a; Khosroshahi, Wadadekar & Kembhavi 00b; öllenhoff & Heidt 01; Graham & Guzmán 03; Aguerri et al. 04; Ravikumar et al. 06; Barway et al. 09; Kourkchi et al. 12). The relation is simply the result of a fitting procedure and is based on the formula (10) for the Sérsic surface brightness profile. A self-correlation between µ 0,b and n follows from the fact that µ is independent on n that is well shown for G09 sample in Fig. 10a. Bulges, pseudos, and elliptical galaxies do not show any trend between µ and n. Such an independence transforms into the linear pseudorelation (Fig. 10b) between µ 0,b and n with a scatter that reflects the range of µ in the samples under discussion. If we use log(n) instead of n, we obtain a curved pseudorelation (Fig. 7). The nature of the curvature in Fig. 6 is exactly the same. The PhP1 includes µ 0,b which according to (14) comprises n. We can approximate n log(n) in a limited range of n and obtain a nearly flat photometric plane in the form (15). In the wider range of n it is curved (Fig. 7) because the relation between µ 0,b and n is linear (Fig. 10). In the next section, we show that the parameter r involved in the relation (15) does not affect our conclusion Effective radius vs Sérsic index for s and ellipticals The existence of the univariate correlation between r and n that might diminish the scatter in the bivariate relation, is very doubtful. Some authors revealed a correlation between r and n (Caon, Capaccioli & D Onofrio 1993; Graham et al. 1996; Guttiérrez et al. 04; öllenhoff 04; La Barbera et al. 04, 05). Khosroshahi, Wadadekar & Kembhavi (00b) and öllenhoff & Heidt (01) give the linear correlation coefficient for this correlation to be ρ > 0.6 with a significance level of %. éndez-abreu et al. (10) were less enthusiastic about this correlation and estimated ρ = 0.28 for their sample of S0-Sb galaxies in the J band. Aguerri et al. (04) analyzed the photometry of 116 bright galaxies from the Coma cluster and found the relation between r and n to be statistically insignificant (ρ = 0.46, P = 0.07). Ravikumar et al. (06) demonstrated that a plot of the Sérsic index against the effective radius shows the presence of two broad distributions (for Es and bright s, for des and faint s of S0s and spirals), but without a good correlation within each group. Barway et al. (09) found systematic differences between bright and faint lenticulars with respect to the Sérsic index as a function of the effective radius. Bright lenticulars are well correlated

12 12 A.V. osenkov, N.Ya. Sotnikova, and V.P. Reshetnikov Figure 9. Distributions over parameters r, µ, and n for the G09 sample. The division into three separate subsamples as it is found in G09. The black dashed line corresponds to faint s, the gray filled histogram is plotted for bright s, and the red solid line corresponds to elliptical galaxies. Vertical lines represent values of r = 0.9 kpc and n = 2.5 to discriminate subsamples with bright and faint s. Figure 10. a) The dependence between the effective surface brightness µ of the underlying host galaxy or of the elliptical and the Sérsic index n; b) the linear relation between the central surface brightness µ 0,b and the Sérsic index n. The data were taken from G09 (r band). Black filled circles represent bright s ( < 19.3 mag), black open circles represent faint s ( 19.3 mag), and red crosses correspond to elliptical galaxies. The solid line corresponds to the median value µ.2 mag arcsec 2 for the subsample of s and pseudos, the dash-dotted line corresponds to the median value µ mag arcsec 2 for the subsample of ellipticals. (ρ = 0.79 with significance greater than %), but faint lenticulars do not show any correlation. G09 sorted out the question about the correlation between r and n. He demonstrated that systems with larger n tend to be more extended but this tendency is rather weak. The Sérsic index n does slowly rise with r for s, but it is rather constant for ellipticals. We replotted these correlations for the sample by G09 in the r band. Fig. 11 represents the plane r n. One can see that objects from three groups (faint s, bright s, and ellipticals) occupy quite different areas in Fig. 11. Bulges and elliptical galaxies are almost perpendicular to each other. For s (rather faint objects) the scatter of r is small. On the contrary, the range of n is large. Thus, the effective radius is almost independent on n. At the same time, bright galaxies (ellipticals) barely show the trend of r with n along the wide area that is almost perpendicular to the n axis. An inhomogeneous sample containing the random mixture of s and ellipticals, faint and bright objects can produce the false correlation between r and n. Additionally, we have gathered data from the samples listed in Table 2 and plotted r against n in the B and r bands (Fig. 12). The result is very convincing. One can notice a slight trend only for small samples, but the overall picture does not show any correlation. Here, it should be noted that we might be missing a significant part of galaxies with small s. The selection effects may put a low limit on the distribution of r which may change the view of found (or unfound) correlations.

13 irages in galaxy scaling relations 13 However, in this paper we do not consider the inhomogeneity or the completness of the samples. Thus, the effective radius is not a third parameter that can improve the relation (15). oreover, in the relation (15) r is expressed in kpc, so the term with log(r ) can give a very small contribution in the x-axis expression presented in Fig. 6, and the leading relation in the expression (15) is a self-correlation (16). 4.5 Photometric Plane 1 as a self-relation To demonstrate the insignificance of the contribution of the term with r which inputs only noise in the relation (), we constructed the PhP1 for the G09 sample (Fig. 13a). For the overall sample we fitted the expression for the PhP1 as: log(n) = log(r ) µ 0,b () Figure 11. The dependence between the effective radius and the Sérsic index plotted for the sample by G09 in the r band (symbols as in Fig. 10). Figure 12. The effective radius versus the Sérsic index for a) the B band; b) the r band. The samples used are listed in Table 2. In the relation () the contribution of the term with r even for large galaxies (log(r ) 0.5) is smaller than the scatter due to µ 3 m (see Fig. 9). Using (16), we rewrote the relation () in the form log(n) = log r µ ν n+1.11, (19) where r and µ are the median values of the effective radius and of the effective surface brightness, respectively. Now one can see that the PhP1 is simply another representation of the self-correlation (14), and the curvature of the PhP1 just shows the curvature of the expression for µ 0,b via log(n). The following trick helps to prove our main idea. We replotted the PhP1 (Fig. 13b) with the same expression () but mixed r values. As it has been done earlier, we split up the sample into three groups of galaxies: faint s (with r 19.3), bright s (with r < 19.3), and ellipticals. The scatter in Fig. 13b increased, mainly for bright and large galaxies with a wide range of r, but the overall shape of the dependence did not change. The curvature of the relation remains the same because the leading and trivial relation (self-correlation) between µ 0,b and log(n) is curved. Ravikumar et al. (06) and Barway et al. (09) noticed that different objects (ellipticals, s, faint and bright galaxies) form different photometric planes with different thicknesses. They noted that ellipticals and s of bright lenticulars have a tight Photometric Plane (PhP1), and they connected this fact with processes that lead to relaxed objects. Now we can see that the main difference in the PhP1s for samples used in our analysis, is the difference in the median value of µ which shifts the plane. The range of µ defines the thickness of the plane. Thus, there is no mystery in the existence of the PhP1 which is simply a self-correlation contaminated by the term r. In Fig. 6 the coefficient under the term µ 0,b is about the same ( ) in different bands while the coefficient under the term r varies substantially. This proves that the leading relation in the expression (15) is the linear dependence of µ 0,b on n with the proviso that µ is independent on n. But two intriguing questions remain. Why is the range of µ for different objects rather small (on average, not greater than 5 m while the luminosity can change up